It is that exciting time of the year where some lucky few in the mathematical community get to choose a graduate school to go to. This post is for you guys. Here are my assumptions: You applied to a bunch of graduate schools and you got into a slightly smaller bunch of graduate schools. Now you think you have a problem: you have to choose one.

The first thing to realize is that this isn’t a problem at all. Very likely any choice you make is as good as any other: you are you no matter where you go. It is (in my opinion) a great privilege to be able to spend time doing math and your time in grad school is going to be perhaps the period in your life where you have the most time to do math ever. It is going to be wonderful!

On the other hand, the choice you make will likely have an enormous impact on what the rest of your life looks like. It will determine who your friends are, where you live, what you eat, etc, etc. Being a graduate student will put a new kind of psychological pressure on you and your time as a graduate student will some sometimes be horrible.

In other words, the choice you make will have an important impact on your life outside of math and I think actually that those consequences are possibly more important than the purely mathematical ones.

Before we get to deciding which school to go to, let’s think about what you will do when you get there: you will write a thesis with a thesis advisor. (To me, as an advisor, this is the only thing that matters.) In the first year or so, besides learning new material, you will choose(!?) your advisor. How will this happen? If you think about what math you understand best, then it is probably the material from the math lectures you liked most. Very likely you will end up working with the professor whose lectures you enjoy the most. I say there is no way of predicting how this will end up and I claim that it is best to go into grad school with no preconceived notion of what will happen.

Having said this, here is the Carpe Diem method of choosing a grad school:

1. Go somewhere else; try something new! Don’t become a graduate student at the institution you are an undergrad at.
2. If at all possible, visit the schools you got into. Talk to the graduate students there, attend a random lecture, and generally just soak in the atmosphere.
3. Try not to worry about extraneous issues like: stipend, teaching, housing, etc. (Of course you may have to for some reason.)
4. Don’t worry about availability of professors. You’ll find somebody to work with, but as I said above there is no telling how, when, why this will happen. If an institution has a certain track record of excellence, you can be sure this will continue in the near future. Moreover, once you are in grad school, the institution you are at has a certain responsibility to get you a PhD (provided you work hard, pass your general exam, etc, etc).
5. Finally, make your choice based on where you think you will enjoy living and working the most.

Just today I finally managed to fix the proof of “formally smooth + locally of finite presentation <=> smooth” for morphisms of algebraic spaces, see Lemma Tag 04AM. In fact, the implication “=>” isn’t hard, and is the result that is used in practice. In the current implementation, the proof of “<=” uses infinitesimal deformation of maps, and in particular a topos theoretic description of first order thickenings of algebraic spaces which we alluded to in this post, see Lemma Tag 05ZN and Lemma Tag 05ZN.

Here is a related fact:

Suppose that X —> Y —> Z are morphisms of algebraic spaces or schemes, that X —> Y is etale and that X —> Z is formally smooth. Then Y —> Z is formally smooth too. 

In other words, being formally smooth is etale local on the source and target. See Lemma Tag 061K for a more precise statement.

If X, Y, Z are schemes, then one can prove this by reducing to the affine case, using that formal smoothness is equivalent to the cotangent complex being a projective module in degree 0 [Edit 5/18/2011: Wrong! See here.], and using the distinguished triangle of cotangent complexes associated to a pair of compose-able ring maps.

If X, Y, Z are algebraic spaces, then one has to do a bit more work (I think). The proof of the reference above uses the material mentioned in the first paragraph and that Ω_{X/Z} is a locally projective, quasi-coherent O_X-module (see Lemma Tag 061I), which is fun in and of itself.

Here is a technically straightforward manner in which to introduce various categories of sheaves on algebraic stacks, and it is my intention to introduce sheaves on algebraic stacks in the stacks project along these lines. Please take a look and leave a comment if you see a problem with this approach.

Suppose that C is a site. Using conventions as in the stacks project:

1. If p : X —> C is a stack in groupoids over C, then we declare a family of morphisms {x_i —> x}_{i ∈ I} in X to be a covering if and only if {p(x_i) —> p(x)}_{i ∈ I} is a covering of the site C. In this way X becomes a site.
2. If f : X —> Y is a 1-morphism of stacks in groupoids over C, then f is a continuous and cocontinuous as a functor of sites. Hence f induces a morphism of topoi f : Sh(X) —> Sh(Y) with the property that the pull back of a sheaf G on Y is defined by the simple rule (f^{-1}G)(x) = G(f(x)). This construction is compatible with composition of 1-morphisms of stacks in groupoids.
3. Finally, if a : f —> g is a 2-morphism in the 2-category of stacks in groupoids over C, then a induces a 2-morphism a : f —> g in the 2-category of topoi.

In other words, this is a perfectly reasonable way to associate a site to each and every stack over C.

Next, let C = (Sch/S)_{fppf} be the category of schemes with the fppf topology as in the stacks project. An algebraic stack X is a category fibred in groupoids over C. Hence the construction above gives us a site X_{fppf} which we will call the fppf site of X. According to the remarks above this has a suitable 2-functoriality with regards to morphisms of algebraic stacks.

Variants: If X is an algebraic stack, then p : X —> C is also a stack fibred in groupoids over C endowed with the Zariski, smooth=etale (see this post), or syntomic topology. Hence we obtain variants X_{Zar}, X_{smooth}, and X_{syntomic} satisfying functorialities as above. Note that the underlying category is X in each case.

Here are some (I think) properties of these definitions:

1. if x is an object of X with U = p(x), then X_{fppf}/x is equivalent (as a site) to (Sch/U)_{fppf}. Hence given a sheaf F on X_{fppf} the cohomology groups H^p(x, F) are just fppf cohomology groups of some sheaf on (Sch/U)_{fppf}. This also works with the other topologies.
2. when the topology is etale=smooth or Zariski, then H^p(x, F) can be computed on the small etale or Zariski site of U.
3. In general X does not have a final object and does not have fibre products. If the diagonal of X is representable (by schemes) then X has all fibre products.
4. Assume the diagonal of X is representable. Let x_0 be an object of X such that U_0 = p(x_0) is a scheme surjective, flat, locally of finite presentation over X. The representable sheaf h_{x_0} surjects onto the singleton sheaf * in Sh(X_{fppf}). Moreover, the fibre products h_{x_0} \times_{*} h_{x_0}, h_{x_0} \times_{*} h_{x_0} \times_{*} h_{x_0}, etc are representable by x_1, x_2, etc with p(x_1) = U \times_X U, p(x_2) = U \times_X U \times_X U, etc. It follows formally from this (compare with Lemma Tag 01GC and Lemma Tag 01GY) that there is a spectral sequence E_1^{p, q} = H^q(x_p, F) => H^{p + q}(X_{fppf}, F) and by the above H^q(x_p, F) corresponds to fppf cohomology of F over the scheme U_p.
5. There is a similar spectral sequence for the smooth=etale topology if the morphism U_0 –> X is surjective and smooth and the diagonal of X is representable.
6. If X is general there is still a spectral sequence with E_1^{p, q} = H^q(U_p, F), but then the U_p are algebraic spaces.

The sheaf of differentials Ω_{X/S} of one scheme X over another scheme S is the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S}. I remember being surprised to learn that people habitually define this sheaf using the conormal sheaf C_{X/Xx_SX} of the diagonal morphism of X over S^[1].

Why is it not the “right thing” to do? The reason is that both the conormal sheaf and the sheaf of differentials have a natural functoriality, and that the identification of C_{X/Xx_SX} with Ω_{X/S} is not compatible with this! Namely, consider the morphism that flips the factors on Xx_SX. This should clearly act by -1 on the conormal sheaf C_{X/Xx_SX} and by +1 on Ω_{X/S}. So there you go!

When X —> S is a morphism of algebraic spaces, then the diagonal morphism isn’t an immersion in general so the conormal sheaf is harder to define. In this case defining Ω_{X/S} as the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S} on the small etale site of X works fine, see Tag 04CR.

Finally, suppose that X —> S is a morphism of algebraic stacks. We have yet to choose (in the stacks project) which site to use to define quasi-coherent sheaves on X. But in order to study differentials the only reasonable choice seems to be the lisse-etale site X_{lisse, etale}. Again there is a universal O_{S_{lisse, etale}}-derivation d : O_{X_{lisse, etale}} —> Ω. Now, (I think) Ω is not a quasi-coherent O_{X_{lisse, etale}}-module, and it is not what authors on algebraic stacks define as Ω_{X/S}, but for some purposes it might be the right thing to look at (e.g., deformation theory?).

Footnote 1: Yes, currently the stacks project also introduces sheaves of differentials for morphisms of schemes using this method. The first result is then that d_{X/S} is a universal derivation, see Lemma Tag 01UR. Having proven this, maps involving Ω_{X/S} are defined using the universal property.